Quantum state tomography and 2-designs

In 1989, Wootters and Fields showed that mutually unbiased bases (MUBs) are examples of optimal orthogonal measurements for quantum state tomography. As Roetteler and Klappenecker observed, MUBs are examples of complex projective 2-designs. In 2006, Scott showed that a general rank-one measurement is optimal for quantum state tomography if and only if it is a 2-design; this result is also true for orthogonal rank-one measurements and "two-outcome" measurements. In this talk, we explain the connection between designs and measurements, and we describe some new constructions and important open problems.